Objective:After studying this section, you will be able to use several methods to prove triangles are similar.

8.3 Methods of Proving Triangles Similar

Postulate If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the three angles of another triangle, then the triangles are similar. (AAA)

Theorem If there exists a correspondence between the vertices of two triangles such that the two angles of one triangle are congruent to the two corresponding angles of other, then the triangles are similar. (AA)

Given:

Prove:

A

B C

D

E F

EB

DA

~ ABC DEF

Theorem If there exists a correspondence between the vertices of two triangles such that the ratios of themeasures of corresponding sides are equal, then the triangles are similar. (SSS similarity)

Given:

Prove:

A

B C

D

E F

DF

AC

EF

BC

DE

AB

~ ABC DEF

Theorem If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs ofcorresponding sides are equal and the included angles are congruent, then the triangles are similar. (SAS Similarity)

Given:

Prove:

A

B C

D

E FEBEF

BC

DE

AB

~ ABC DEF

Given: parallelogram YSTW

Prove:

D

BA

C

F

E

~ BFE CFD

Given:

Prove: ~ PEN PAL

EARP

N

EALP

trisect and

LP ofmidpoint theis

E P R A

L

N

Given:

Prove: ~ KHJ HGJ

GHJright of GJ hypotenuse

toaltitude theis

KH

G

J

H

K

The sides of one triangle are 8, 14, and 12, and the sides of another triangle are 18, 21, and 12. Prove that the triangles are similar.

Summary: If you were to draw a triangle and have a line parallel to the base,creating two triangles, would the triangles be similar if you had a right triangle? Obtuse? Acute? Why?

Homework: worksheet